Download Section6

Chapter 2
Section 6
Complementary and Supplementary Theorems
and Planning a Proof
5 parts to proving a theorem
1) Statement of the theorem
2) Diagram illustrating the given information
(or hypothesis)
3) A list of the given (hypothesis)
4) A list of what is to be proven (conclusion)
5) A series of conditionals with their reasons
in logical order that lead from the given to
the prove (the proof!!)
Supplementary R‘s Theorem
• If two angles are supplements of congruent angles (or
the same angle),then the two angles are congruent
Complementary R‘s Theorem
• If two angles are complements of congruent angles (or
the same angle),then the two angles are congruent
Proof of Supplementary Theorem
•
If two angles are supplements of congruent angles (or the same angle),then the two angles are
congruent
Given:
<1 and <2 are supp
<3 and <4 are supp
1
<2 @ <4
Prove:
2
3
4
<1 @ <3
Hypothesis
conclusion
reason
1) None
<1 and <2 are supp
given
<3 and <4 are supp
<2 @ <4
2) If <1 and <2 are supp
then m<1 + m<2 = 180
and <3 and <4 are supp
3) If m<1 + m<2 = 180
m<3 + m<4 = 180
o
and m<3 + m<4 = 180
Def supplementary angles
o
o
then m<1 + m<2 = m<3 + m<4
Transitive / substitution prop
then m<1 = m< 3 or <1 @ <3
Subtraction property
o
4) If m<1 + m<2 = m<3 + m<4
m<2 =
(or m<2 = m< 4)
m< 4
This is a test question :
True or false??
If R A is complementary to R B and R B is complementary to R C
then R A is complementary to R C
False!!
Counter example: let m R B = 30 then A & C’s sum is 120
When is this true?
o
When the measure of R B is 45
o
Practice work
• P63 we 1-17 all
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
definitions
An example used to prove an if-then statement is false
Another name for a statement written in the if then form
The proposition being investigated
A conditional in which the hypothesis and conclusion
have been interchanged
Reasoning proven using given information, definitions,
postulates, and theorems already proven
An adjacent pair of angles whose exterior sides are
perpendicular
A statement that contains the words “if and only if”
An adjacent pair of angles whose exterior sides are
opposite rays
Two angles such that the sides of one angle are opposite
rays to the sides of the other angle
The statement in an argument that follows as a result of
the hypothesis